LIMIT THEOREMS FOR RANDOM CUBICAL HOMOLOGY YASUAKI HIRAOKA AND KENKICHI TSUNODA ABSTRACT. This paper studies random cubical sets in Rd. Given a cubical set X ⊂ Rd, a random variable ωQ ∈ [0, 1] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ ≤ t for each t ∈ [0, 1]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown. 1. INTRODUCTION The mathematical subject studied in this paper is motivated by imaging science. Objects in Rd are usually represented by cubical sets, which are the union of pixels or higher dimensional voxels called elementary cubes, and those digitalized images are used as the input for image processing. For example, image recognition techniques identify patterns and characteristic shape features embedded in those digital images. Recent progress on computational topology [6, 13] allows us to utilize homology as a descriptor of the images. Here, homologyis an algebraic tool to study holes in geometricobjects, and it enables us to extract global topological features in data (e.g., [1, 15, 16]). In particular, the mathematical framework mentioned above is called cubical homology, which will be briefly explained in Subsection 2.1 (see [13] for details). In applications, digital images usually contain measurement/quantization noise, and hence it is important to estimate the effect of randomness on cubical homology. Fur- thermore, by studying the asymptotic behaviors of randomized cubical homology, we can understand the (homological) structures in the images as a difference from the random states. Random topology is a mathematical subject to study these problems, and it is a new branch of mathematics that has emerged in the intersection between algebraic topol- ogy and probability theory. The reader may refer to the survey papers [2, 14] and the arXiv:1612.08485v2 [math.PR] 19 Mar 2018 references therein for further details. In this paper, we study several limit theorems for random cubical homology. Our model, which will be precisely explained in Subsection 2.2, assigns a random variable ωQ on [0, 1] from a probability measure P for each elementary cube Q in a cubical set X ⊂ Rd. Then, for each t, we construct a random cubical set as a sublevel set X(t)= {Q ∈ X : ωQ ≤ t}. This is a natural higher dimensional generalization of the classical bond percolation model [9] in which the elementary cubes only consist of vertices and edges in Zd. This model also contains the Bernoulli random cubical complex [12], which is an analog of the Linial- Meshulam random simplicial complex [17]. Furthermore, the sequence {X(t)}0≤t≤1 can 2010 Mathematics Subject Classification. 60D05, 52C99, 60F05, 60F15. Key words and phrases. Random topology, Cubical complex, Cubical homology, Betti number. 1 2 YASUAKIHIRAOKAANDKENKICHITSUNODA be regarded as a random filtration and this naturally connects to the concept of persistent homology [7, 22]. Under this setting, we study the law of large numbers (LLN) and the central limit the- orem (CLT) for the Betti numbers βq(t) := βq(X(t)). We also study the asymptotic behavior of the lifetime sum of persistent homology. Here, given a filtration such as {X(t)}0≤t≤1, the q-th persistent homology characterizes each q-dimensional hole c by encoding its birth and death parameters t = b, d (0 ≤ b ≤ d ≤ 1), where b and d express the appearance and disappearance of the hole c, respectively. Then, the lifetime sum Lq is defined by the sum of the lifetime d − b for all q-dimensional holes. Alternatively, it can 1 be also expressed by the integral Lq = 0 βq(t)dt of the Betti number. The lifetime sum has a natural connection to the classical theorem in combinatorial probability theory called Frieze’s ζ(3) theorem [8], and a higherR dimensional generalization of Frieze’s theorem is studied in [10, 11, 12]. In view of this connection, we also prove the LLN and the CLT for the lifetime sum of the persistent homology in this paper. We remark on several papers related to this work. First, the paper [12] studies the Bernoulli random cubical complex and determines the asymptotic order of the expected lifetime sum of the persistent homology. Hence, the LLN for the lifetime sum in this paper is stronger than the result in [12]. We also remark that the paper [20] discusses several random cubical sets, and gives exact polynomial formulae for the expected value and variance of the intrinsic volumes. The study on cubical homology is addressed in the future work in [20]. Instead of discrete settings, the limit theorems for Betti numbers defined on point processes in Rd are studied in [21]. The paper [5] also shows the limit theorems for persistence diagrams on point processes in Rd. The paper is organized as follows. In Section 2, after brief introduction of cubical sets and cubical homology, we explain our model of random cubical sets and state the main results of the LLN and the CLT for Betti numbers and lifetime sums. Some computations of limiting Betti numbers are also presented at the end of this section. The proofs for the LLN and the CLT are given in Section 3 and 4, respectively. At the end of Section 3, we also show a sufficient condition for the positivity of the limiting Betti numbers. Section 5 concludes the paper and shows some future problems. 2. MODEL AND MAIN RESULTS 2.1. Cubical homology. We review in this subsection the concept of cubical homology, which is a main object studied in this paper. This subsection is devoted to a brief summary of Chapter 2 in [13]. We refer to [13] for more detailed description and the proofs of propositions introduced in this subsection. An elementary interval is a closed interval I ⊂ R of the form I = [l,l + 1] or I = [l,l] , for some l in Z. In the latter case, we shall write it as I = [l]. Fix d ∈ N. Throughout the paper, d represents the dimension of the state space where cubical sets, which will be defined later, are considered. An elementary cube Q is a finite product of elementary intervals of the form Q = I1 ×···× Id , where Ii ⊂ R is an elementary interval for each 1 ≤ i ≤ d. Let Q be an elementary cube of the form Q = I1 ×···× Id with elementary intervals Ii, 1 ≤ i ≤ d. For each 1 ≤ i ≤ d, denote by Ii(Q) the i-th component of Q: Ii(Q)= Ii. In this paper, the term “filtration” means an increasing family of cubical complexes as usual in topology. LIMITTHEOREMSFORRANDOMCUBICALHOMOLOGY 3 For an elementary interval I ⊂ R, I is said to be nondegenerate if I is not a singleton. We denote by dim Q the number of nondegenerate components of Q: dim Q = #{1 ≤ i ≤ d : Ii(Q) is nondegenerate} . Define Kd by the set of all elementary cubes in Rd. For each 0 ≤ k ≤ d, we also define d d Kk as the set of all elementary cubes in R whose dimension is equal to k. A subset X ⊂ Rd is said to be cubical if X can be written as a union of elementary cubes in Rd. Note that a cubical set is a subset of Rd with this definition, not a set of chains (see also Remark 2.3). Note that an infinite union of elementary cubes in Rd is also included in our definition of cubical sets although it is assumed to be a finite union in [13]. For a cubical set X ⊂ Rd, denote by Kd(X) the set of all elementary cubes contained in d X. For each 0 ≤ k ≤ d, we also denote by Kk(X) the set of all elementary cubes in X whose dimension is equal to k. d For each elementary cube Q in Kk, let Q be an algebraic object of Q. Q is called an d elementary k-chain. Denote by Kk the set of all elementary k-chains. We define the Z-free d d b b module Ck = Z(Kk) by the module over Z generated by all elementary k-chains: b d d b Ck := {c = αiQi : Qi ∈ Kk, αi ∈ Z} . finite sum X d b b b d An element belonging to Ck is called a k-dimensional chain. We also set Ck := 0 for d d k< 0 or k > d. For a cubical set X ⊂ R , we similarly denote by Kk(X) the set of all el- d d ementary k-chains in X and define the Z-free module Ck (X)= Z(Kk(X)), respectively: b d d Kk(X) := {Q : Q ∈ Kk(X)} , b Cd(X) := {c = α Q : Q ∈ Kd(X), α ∈ Z} . bk b i i i k i finite sum X bd b b m m Consider k-dimensional chains c1,c2 ∈ Ck with c1 = i=1 αiQi and c2 = i=1 βiQi. The scalar product of the chains c1 and c2 is defined as P P m b b hc1,c2i = αiβi . i=1 X ′ d d For elementary cubes P ∈ Kk and Q ∈ Kk′ , we define the cubical product P ⋄ Q by \ P ⋄ Q := P × Q. b b ′ d d Note that, for elementary cubes P ∈ K and Q ∈ K ′ , the direct product P × Q is also an b′ kb k d+d elementary cube belonging to Kk+k′ .